EuAl4: CDW and Magnetic Phase Complexity

• EuAl4 is a rare-earth intermetallic featuring a BaAl4-type structure that hosts a transverse incommensurate CDW with non-centrosymmetric modulations.
• It undergoes a cascade of magnetic transitions below 15.4 K, manifesting multiple antiferromagnetic states along with field-induced skyrmion and other topological spin textures.
• The interplay of electron–phonon coupling, Fermi-surface nesting, and symmetry breaking provides key insights into the complex coupling between lattice, charge, and magnetic dynamics.

EuAl$_4$ is a BaAl$_4$-type rare-earth intermetallic whose low-temperature physics is governed by the coexistence of an incommensurate charge-density wave (CDW), multiple antiferromagnetic phases, and field-induced topological spin textures. At room temperature it crystallizes in the tetragonal space group $I4/mmm$; below $T_{\mathrm{CDW}} \approx 145$ K it develops a long-wavelength modulation with $\mathbf{q}\parallel \mathbf{c}^{*}$, and below $T_N=15.4$ K it enters a cascade of magnetically ordered states. A central result of recent diffraction work is that the CDW phase is non-centrosymmetric, with orthorhombic superspace symmetry $F222(0\,0\,\sigma)00s$, which directly affects microscopic interpretations of the skyrmion phases and of the coupling between lattice, charge, and magnetism (Kotla et al., 2 Jun 2025).

1. Crystal chemistry and baseline phase diagram

EuAl$_4$ is a tetragonal intermetallic of the BaAl$_4$ structure type, space group $I4/mmm$ (No. 139), with three crystallographically independent atoms: Eu, Al1, and Al2. At 160 K the lattice parameters are $_4$0, $_4$1, and $_4$2 (Kotla et al., 2 Jun 2025). Eu is divalent, $_4$3, with localized $_4$4 moments and $_4$5, while the itinerant states relevant to transport and the CDW are derived primarily from the Al network (Kotla et al., 2 Jun 2025).

EuAl$_4$6 is metallic and has repeatedly been discussed as a topological semimetal or topological magnet. In the Eu(Ga$_4$7Al$_4$8)$_4$9 series, EuAl$I4/mmm$0 is the Al-rich end member and one of the compositions that exhibits a clear CDW-like transport anomaly at ambient pressure; the zero-field resistivity is metallic and the residual resistivity ratio is approximately 70, consistent with high crystal quality (1804.02076).

Regime	Characteristic temperature or vector	Established feature
CDW onset	$I4/mmm$1 K	Incommensurate modulation with $I4/mmm$2
Magnetic ordering onset	$I4/mmm$3 K	First zero-field antiferromagnetic transition
Additional zero-field magnetic transitions	$I4/mmm$4 K, $I4/mmm$5 K, $I4/mmm$6 K	Cascade of distinct magnetic phases
Field-induced textures for $I4/mmm$7	$I4/mmm$8, $I4/mmm$9	Single-$T_{\mathrm{CDW}} \approx 145$0 spirals, rhombic and square skyrmion lattices, vortex and meron phases

The magnetic nomenclature depends on the probe. In one convention, zero-field cooling passes through PM $T_{\mathrm{CDW}} \approx 145$1 VII $T_{\mathrm{CDW}} \approx 145$2 VI $T_{\mathrm{CDW}} \approx 145$3 V $T_{\mathrm{CDW}} \approx 145$4 I, while field along $T_{\mathrm{CDW}} \approx 145$5 drives I $T_{\mathrm{CDW}} \approx 145$6 II $T_{\mathrm{CDW}} \approx 145$7 III $T_{\mathrm{CDW}} \approx 145$8 IV $T_{\mathrm{CDW}} \approx 145$9 forced FM; phases II and III are rhombic and square skyrmion lattices, respectively (Gen et al., 10 Nov 2025). Resonant magnetic x-ray scattering resolves the four zero-field ordered states as AFM1–AFM4, each with single-$\mathbf{q}\parallel \mathbf{c}^{*}$0 incommensurate order (Vibhakar et al., 2024).

2. Charge-density wave: modulation, transverse character, and phason disorder

Below $\mathbf{q}\parallel \mathbf{c}^{*}$1, satellite reflections appear in single-crystal diffraction with modulation vector $\mathbf{q}\parallel \mathbf{c}^{*}$2. Representative refinements give $\mathbf{q}\parallel \mathbf{c}^{*}$3 at 70 K, $\mathbf{q}\parallel \mathbf{c}^{*}$4 at 20 K, and $\mathbf{q}\parallel \mathbf{c}^{*}$5 at 30 K (Ramakrishnan et al., 2022, Kotla et al., 2 Jun 2025). No splitting or broadening of the fundamental Bragg peaks is observed in the CDW regime, so the lattice remains metrically tetragonal even though the full modulated structure has lower symmetry (Kotla et al., 2 Jun 2025).

The modulation is transverse. For $\mathbf{q}\parallel \mathbf{c}^{*}$6, the dominant atomic displacements are within the $\mathbf{q}\parallel \mathbf{c}^{*}$7 plane, so that $\mathbf{q}\parallel \mathbf{c}^{*}$8. In superspace parameterization this is expressed through first-order harmonic displacement functions,

$\mathbf{q}\parallel \mathbf{c}^{*}$9

with the nonzero coefficients concentrated in in-plane components (Kotla et al., 2 Jun 2025). Neutron Laue diffraction had already shown that the CDW superlattice peaks are absent along the $T_N=15.4$0 axis, which is consistent with a modulation mainly due to in-plane displacements of Al ions rather than longitudinal displacements along $T_N=15.4$1 (Kaneko et al., 2021).

A further refinement of the structural picture is the identification of phason disorder. In the 30 K single-crystal x-ray data, first-harmonic displacement modulation alone systematically overestimates the intensities of second-order satellites, producing the “$T_N=15.4$2 problem.” Introducing second-order harmonic modulation of the anisotropic displacement parameters resolves this discrepancy, whereas higher-harmonic displacement modulation does not. This is taken as the hallmark of phason dynamics in an incommensurate structure, so the CDW in EuAl$T_N=15.4$3 is best described as a transverse CDW with significant phason disorder (Kotla et al., 2 Jun 2025).

Real-space cryogenic 4D-STEM imaging is consistent with this description but emphasizes the internal structure of the modulation. It resolved two out-of-phase intra-unit-cell shear modulations with wavelength $T_N=15.4$4 nm, one associated with Al1–Al2 distortions and one with a $T_N=15.4$5-shear of the unit cell, showing directly that the long-wavelength CDW carries internal degrees of freedom beyond a single scalar amplitude (Ni et al., 2023).

3. Superspace symmetry and the inversion-symmetry problem

The symmetry of the CDW phase has been a central issue because earlier probes supported different superspace descriptions. Single-crystal x-ray diffraction first established an orthorhombic CDW on the tetragonal lattice and assigned the superspace group $T_N=15.4$6, with the fourfold symmetry broken entirely by the modulation wave (Ramakrishnan et al., 2022). Subsequent inelastic x-ray scattering and lattice-dynamics work argued that the soft-mode eigenvector is most naturally described by $T_N=15.4$7 (Korshunov et al., 2024). Cryogenic 4D-STEM then showed that the modulation breaks inversion symmetry locally while preserving it on average, yielding local point groups compatible with non-centrosymmetric environments (Ni et al., 2023).

The decisive structural refinement uses second-order satellites. At 30 K the synchrotron dataset contains 207 unique main reflections, 380 unique first-order satellites, and 394 unique second-order satellites, of which 31 are observed above $T_N=15.4$8. Refinements over six candidate superspace groups show that the best agreement is obtained for the non-centrosymmetric orthorhombic superspace group

$T_N=15.4$9

with $F222(0\,0\,\sigma)00s$0 and $F222(0\,0\,\sigma)00s$1. The centrosymmetric alternatives $F222(0\,0\,\sigma)00s$2 and $F222(0\,0\,\sigma)00s$3 fit the second-order satellites significantly worse (Kotla et al., 2 Jun 2025).

In this description the average lattice remains essentially tetragonal, but the modulation lowers the symmetry from $F222(0\,0\,\sigma)00s$4 to $F222(0\,0\,\sigma)00s$5. The transformation from the tetragonal $F222(0\,0\,\sigma)00s$6-cell to the orthorhombic $F222(0\,0\,\sigma)00s$7-cell is

$F222(0\,0\,\sigma)00s$8

The loss of inversion is accompanied by site splitting, notably Al1 $F222(0\,0\,\sigma)00s$9 Al1a + Al1b, and by symmetry-allowed differences in the modulation functions of those sites (Kotla et al., 2 Jun 2025).

This resolves a long-running controversy. The CDW phase is not merely orthorhombic in a centrosymmetric sense; it is acentric in the full superspace description. A plausible implication is that local inversion breaking observed in microscopy and average acentricity established by diffraction are two descriptions of the same structural fact at different levels of resolution (Ni et al., 2023, Kotla et al., 2 Jun 2025).

4. Magnetic order, skyrmion phases, and microscopic interpretations

Below $_4$0 K, EuAl$_4$1 develops multiple incommensurate antiferromagnetic phases. Time-of-flight neutron Laue diffraction found $_4$2 with $_4$3 at 13.5 K, then an abrupt change below $_4$4 K to $_4$5 with $_4$6 at 11.5 K and $_4$7 at 4.3 K (Kaneko et al., 2021). Resonant magnetic x-ray scattering later resolved AFM1 as an in-plane spin-density wave, AFM2 as coexistence of that SDW with a second SDW having moments along $_4$8, AFM3 as a single-chirality magnetic helix, and AFM4 as a helix with reversed chirality; all four phases remain single-$_4$9 (Vibhakar et al., 2024).

A distinctive low-temperature result is the spontaneous reversal of spin chirality. Below $_4$0 K the helix is stabilized with a single chirality across the sample, while below $_4$1 K the chirality reverses and the sample remains a single chiral domain. Concomitantly, the symmetry lowers to polar monoclinic, with uniaxial charge and spin strip domains. Group-theoretical analysis shows that the polar monoclinic symmetry is required to explain the asymmetry of the two chiral states and the chirality reversal (Vibhakar et al., 2024).

Under field $_4$2, EuAl$_4$3 hosts a rhombic skyrmion lattice in phase II, a square skyrmion lattice in phase III, vortex–antivortex phases, meron–antimeron textures, and single-$_4$4 spirals. The fundamental modulation vectors of phases III, VI, and VII are $_4$5, $_4$6 with $_4$7, while phase I carries $_4$8 and phase V $_4$9 (Gen et al., 10 Nov 2025).

The microscopic origin of these textures is actively debated. The structural identification of the CDW phase as non-centrosymmetric $I4/mmm$0 means that ordinary Dzyaloshinskii–Moriya interactions are symmetry-allowed below $I4/mmm$1, so a more exotic mechanism is not required to account for skyrmions in the ordered state (Kotla et al., 2 Jun 2025). By contrast, soft-x-ray ARPES on Eu(Ga$I4/mmm$2Al$I4/mmm$3)$I4/mmm$4 argues that multiple nesting vectors derived from a Z-centered Fermi-surface pocket match the periodicities and symmetries of the helical and skyrmion phases, suggesting a common origin in competing nesting-induced RKKY interactions (Arai et al., 14 Apr 2026). This suggests that realistic models of EuAl$I4/mmm$5 must account simultaneously for symmetry-allowed DM terms and for strongly momentum-selective itinerant exchange.

5. Electronic structure, phonons, and transport renormalization

Band-structure calculations and ARPES consistently place EuAl$I4/mmm$6 in the class of three-dimensional topological semimetals. In the tetragonal basic structure a Dirac nodal crossing occurs above $I4/mmm$7 along $I4/mmm$8–$I4/mmm$9, protected by $_4$00, while the partial density of states at $_4$01 is dominated by Al-derived states and the Eu $_4$02 manifold lies well below $_4$03 (Ramakrishnan et al., 2022). Soft-x-ray ARPES across the Eu(Ga$_4$04Al$_4$05)$_4$06 series further identified a Lifshitz transition between EuGa$_4$07 and EuGa$_4$08Al$_4$09, where a Z-centered electron pocket emerges; in EuAl$_4$10 this pocket supplies the nesting vectors that match the zero-field helix and the square skyrmion lattice (Arai et al., 14 Apr 2026).

The low-temperature electronic structure is strongly reconstructed by magnetism. Laser ARPES showed that EuAl$_4$11 undergoes band splitting, backfolding, the appearance of new Fermi sheets, and a large enhancement of quasiparticle lifetime across the AFM transitions, with the most dramatic changes at the AFM3 $_4$12 AFM4 transition rather than at $_4$13. This coincides with the largest drop in resistivity and indicates that the detailed magnetic structure, not merely the presence of order, controls carrier coherence (Eaton et al., 2024).

The origin of the CDW is now tied to momentum-dependent electron–phonon coupling. Inelastic x-ray scattering revealed a broad softening of a transverse acoustic branch along $_4$14–$_4$15 that freezes out at $_4$16, and the eigenvector of that soft mode matches the displacement pattern of the modulated phase. The broad anomaly, together with the absence of a sharply peaked susceptibility, places EuAl$_4$17 in the “type II” category of EPC-driven CDWs rather than a simple Peierls nesting picture (Korshunov et al., 2024). Comparative Wannier-based susceptibility calculations across BaAl$_4$18-type compounds reached the same conclusion: the CDW in EuAl$_4$19 and SrAl$_4$20 requires strong EPC to a transverse acoustic mode at small $_4$21 along $_4$22–$_4$23, in addition to a maximum in $_4$24 (Wang et al., 2023).

Pressure and Raman spectroscopy expose the same hierarchy. High-pressure IXS under diamond-anvil conditions shows that the EPC responsible for the CDW is progressively suppressed by hydrostatic pressure, with $_4$25 K/GPa and a critical pressure $_4$26 GPa for CDW suppression; the phonon self-energy analysis identifies a critical EPC amplitude $_4$27 meV at the transition (Sukhanov et al., 16 Feb 2026). Raman measurements, in turn, found that below $_4$28 K the Fano asymmetry $_4$29 of the $_4$30 and $_4$31 phonons decreases with the free-carrier density, indicating weakened EPC in the CDW ground state, while the $_4$32 linewidth reveals enhanced phonon–phonon interactions and stronger lattice anharmonicity. The same Raman work identified shoulder-like anomalies around 50 K, suggesting a possible intermediate electronic state between the high-temperature metal and the fully developed CDW regime (Cao et al., 4 Jan 2025).

6. Tuning, family relationships, and unresolved directions

EuAl$_4$33 is unusually sensitive to weak symmetry-breaking perturbations. Compressive uniaxial stress along $_4$34 of only several tens of MPa enhances antiferromagnetic character, increases the helix wavevector in phase I from $_4$35 at 0 MPa to $_4$36 at 80 MPa, raises $_4$37 with slope $_4$38 K/GPa, suppresses phase V, and destabilizes the square skyrmion lattice in favor of other phases (Gen et al., 10 Nov 2025). First-principles calculations in that study show that orthorhombic distortion reshapes the Fermi surface and changes the nesting vectors, supporting a direct route from lattice distortion to magnetic modulation.

Within the broader BaAl$_4$39 family, EuAl$_4$40 is closely related to SrAl$_4$41, which also hosts a non-centrosymmetric transverse modulation described by $_4$42, and to EuAl$_4$43Ga$_4$44, whose CDW instead adopts the I-centered orthorhombic superspace group $_4$45. Despite that difference, both EuAl$_4$46 and EuAl$_4$47Ga$_4$48 place the CDW primarily on the Al1-type layers (Ramakrishnan et al., 2023, Agarwal et al., 2024). In the Eu(Ga$_4$49Al$_4$50)$_4$51 series, only $_4$52 and $_4$53 show CDW-like transport anomalies at ambient pressure, which was attributed to the combined effects of chemical order and chemical pressure (1804.02076).

Surface-sensitive probes add another layer. ARPES and STM on EuAl$_4$54 reveal a $_4$55 surface reconstruction with ordered 50% Eu vacancies, quasi-one-dimensional modulations, and unidirectional replica bands orthogonal to the bulk CDW vector; these features disappear irreversibly on thermal cycling, indicating decoupled surface and bulk orders (Li et al., 5 Sep 2025). This establishes that the bulk incommensurate CDW does not exhaust the symmetry-lowering phenomena accessible in EuAl$_4$56.

Several questions remain open. A full symmetry-consistent refinement of all magnetic phases in the non-centrosymmetric CDW background is still required; the role of phason disorder in magnetic pinning and dynamics is unresolved; and the relative weights of CDW-enabled DM interactions and nesting-driven RKKY in stabilizing the multiple skyrmion phases remain under active discussion (Kotla et al., 2 Jun 2025, Arai et al., 14 Apr 2026). What is already clear is that EuAl$_4$57 is not adequately described as a simple tetragonal antiferromagnet with an incidental superstructure: its defining property is the mutual renormalization of a transverse incommensurate CDW, itinerant electronic structure, and unusually elaborate Eu-moment magnetism.